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Comment by Jun8 | Hacker News Reader

Hah, serendipity: I was reading the Groups part of the Infinite Napkin after it was posted on HN recently. I understand the definitions, etc. but still haven’t grasped the central importance of groups.

For example, article says there are 50 groups of order 72 (chatGPT says there are 50 non-Abelian, 5 Abelian), this seems to be an important insight but into what?

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Don't listen to ChatGPT. There are 44 non-abelian and 6 abelian groups of order 72. I wouldn't say these particular numbers are terribly important, but they are correct.

Don't listen to ChatGPT.

As someone "junior programmer" , I think I should just keep this information . Many times I have tried to not use chatgpt. but its just so lucrative.

I really need better control of myself. Going to block chatgpt at a dns level.

Definitely don't use ChatGPT to try to learn about something you have no knowledge of. It's impossible to separate the bullshit from the truth if you don't already have some foundation in the field, and chatbots often try to sneak some bullshit in there.

Groups are important because they are the algebraic way to describe symmetry: if you have some operation that leaves a thing invariant (e.g. rotating an equilateral triangle so that a vertex lands on where another started), then the operation is invertible and the inverse leaves the thing invariant. You can compose such operations and the composition will still be invariant. The identity function always leaves everything invariant. So your symmetry operations form a group.

Slightly trickier is that every group is a set of symmetry operations for something. So groups exactly capture the idea of symmetry. To a mathematician, "group" and "symmetries" are synonymous.

Finite groups can be interesting as the symmetries of e.g. molecules (e.g. rotating atoms around onto each other), which can tell you something about molecular structure, energy levels, spectra, bonding potential, etc. Infinite groups appear in physics (e.g. the laws of physics are the same when you rotate or translate your coordinates by arbitrary amounts). Symmetry also comes up as a way to study other mathematical objects, and mathematicians might just want to know what all possible groups look like.

The presence of ann invariant implies invertibility! That doesn’t seem true.

Right, e.g. projection onto a subspace of a vector space leaves that subspace invariant. I suppose that the existence of an inverse is axiomatic (and maybe up for debate for whether it is required to consider something a "symmetry"), but assuming one exists, then it leaves the object invariant. You could also observe that it has to be invertible on the object, but not obviously necessarily the ambient space I suppose. Representation theory perhaps has a satisfying answer to that part.

I've only taken three (undergrad) classes involving groups so I'm far from an expert but my feeling is that their underlying structure is a bit like prime numbers.

Nobody bothers explaining why the primes are spaced like they are, rather people explain other phenomena by pointing out that certain things about it are prime or not.

For instance, there's this thing about having a nice neat formula for factoring second degree polynomials (the quadratic formula). One also exists for cubics and quartics (though they don't usually have you memorize these) but none exists for quintics. It took mathematicians a while to prove that such a thing doesn't exist (how to prove a negative?) but they managed it by arguing that all such things have an underlying finite group smaller than a certain size and look, we've listed them here, and there's no such group corresponding to a quintic formula, therefore there is no quintic formula.

So they're useful as a sort of primordial complexity that can be referenced without extensive explanation since properties about the small ones can be checked by hand. And as it turns out, quite a lot of things form groups if you bother you look at them that way.

> Nobody bothers explaining why the primes are spaced like they are

I take it that means you also haven't taken (m)any number theory classes then ;-). Because people wildly care about that.

This is in a sense the background of another well-known conjecture, the Riemann conjecture...

Fair point, and correct (just one class). Myself, I'm fine with a field being important because well it's a mystery of the universe and well we just couldn't not scratch the itch. But I know that not all askers want answers like that, so I was attempting to dust off the "it's also useful" neurons :)

Many mathematical structures are groups so understanding them helps you get insight into the concrete situation you are trying to solve. For example is the problem I am seeing abelian? If so then it must look like X or Y.

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